Efficient implementation of a multidimensional fast fourier transform on a distributed-memory parallel multi-node computer

ABSTRACT

The present in invention is directed to a method, system and program storage device for efficiently implementing a multidimensional Fast Fourier Transform (FFT) of a multidimensional array comprising a plurality of elements initially distributed in a multi-node computer system comprising a plurality of nodes in communication over a network, comprising: distributing the plurality of elements of the array in a first dimension across the plurality of nodes of the computer system over the network to facilitate a first one-dimensional FFT; performing the first one-dimensional FFT on the elements of the array distributed at each node in the first dimension; re-distributing the one-dimensional FFT-transformed elements at each node in a second dimension via “all-to-all” distribution in random order across other nodes of the computer system over the network; and performing a second one-dimensional FFT on elements of the array re-distributed at each node in the second dimension, wherein the random order facilitates efficient utilization of the network thereby efficiently implementing the multidimensional FFT. The “all-to-all” re-distribution of array elements is further efficiently implemented in applications other than the multidimensional FFT on the distributed-memory parallel supercomputer.

CROSS REFERENCE

[0001] The present invention claims the benefit of commonly-owned,co-pending U.S. Provisional Patent Application Serial No. 60/271,124filed Feb. 24, 2001 entitled MASSIVELY PARALLEL SUPERCOMPUTER, the wholecontents and disclosure of which is expressly incorporated by referenceherein as if fully set forth herein. This patent application isadditionally related to the following commonly-owned, co-pending UnitedStates Patent Applications filed on even date herewith, the entirecontents and disclosure of each of which is expressly incorporated byreference herein as if fully set forth herein. U.S. patent applicationSer. Nos. (YOR920,020,027US1, YOR920,020,044US1 (15270)), for “ClassNetworking Routing”; U.S. patent application Ser. No. (YOR920,020,028US1(15271)), for “A Global Tree Network for Computing Structures”; U.S.patent application Ser. No. (YOR920,020,029US 1 (15272)), for ‘GlobalInterrupt and Barrier Networks”; U.S. patent application Ser. No.(YOR920,020,030US1 (15273)), for ‘Optimized Scalable Network Switch”;U.S. patent application Ser. Nos. (YOR920,020,031US1, YOR920,020,032US1(15258)), for “Arithmetic Functions in Torus and Tree Networks’; U.S.patent application Ser. Nos. (YOR920,020,033US1, YOR920,020,034US1.(15259)), for ‘Data Capture Technique for High Speed Signaling”; U.S.patent application Ser. No. (YOR920,020,035US1 (15260)), for ‘ManagingCoherence Via Put/Get Windows‘; U.S. patent application Ser. Nos.(YOR920,020,036US1, YOR920,020,037US1 (15261)), for “Low Latency MemoryAccess And Synchronization”; U.S. patent application Ser. No.(YOR920,020,038US1 (15276), for ‘Twin-Tailed Fail-Over for FileserversMaintaining Full Performance in the Presence of Failure“; U.S. patentapplication Ser. No. (YOR920,020,039US1 (15277)), for “Fault IsolationThrough No-Overhead Link Level Checksums‘; U.S. patent application Ser.No. (YOR920,020,040US1 (15278)), for “Ethernet Addressing Via PhysicalLocation for Massively Parallel Systems”; U.S. patent application Ser.No. (YOR920,020,041US1 (15274)), for “Fault Tolerance in a SupercomputerThrough Dynamic Repartitioning”; U.S. patent application Ser. No.(YOR920,020,042US1 (15279)), for “Checkpointing Filesystem”; U.S. patentapplication Ser. No. (YOR920,020,043US1(15262)), for “EfficientImplementation of Multidimensional Fast Fourier Transform on aDistributed-Memory Parallel Multi-Node Computer”; U.S. patentapplication Ser. No. (YOR9-20,010,211US2 (15275)), for “A NovelMassively Parallel Supercomputer”; and U.S. patent application Ser. No.(YOR920,020,045US1 (15263)), for “Smart Fan Modules and System”.

BACKGROUND OF THE INVENTION

[0002] 1. Technical Field of the Invention

[0003] The present invention generally relates to a field ofdistributed-memory message-passing parallel multi-node computers andassociated system software, as applied for example to computations inthe fields of science, mathematics, engineering and the like. Moreparticularly, the present invention is directed to a system and methodfor efficient implementation of a multidimensional Fast FourierTransform (i.e., “FFT”) on a distributed-memory parallel supercomputer.

[0004] 2. Description of the Prior Art

[0005] Linear transforms, such as the Fourier Transform (i.e., “FT”),have widely been used for solving a range of problems in the fields ofscience, mathematics, engineering and the like. The FT alters a givenproblem into one that may be more easily solved, and the FT is used inmany different applications. For example, for a system of N variables,the FT essentially represents a change of the N variables fromcoordinate space to momentum space, where the new value of each variabledepends on the values of all the old variables. Such a system of Nvariable is usually stored on a computer as an array of N elements. TheFT is commonly computed using the Fast Fourier Transform (i.e., “FFT”).The FFT is described in many standard texts, such as the NumericalRecipes by Press, et al. (“Numerical Recipes in Fortran”, pages 490-529,by W. H. Press, S. A. Teukolsky, W. A. Vetterling and Brian P Flannery,Cambridge University Press, 1986, 1992, ISBN: 0-521-43064-X). Mostcomputer manufacturers provide library function calls to optimize theFFT for their specific processor. For example, the FFT is fullyoptimized on the IBM's RS/6000 processor in the Engineering andScientific Subroutine Library. These library routines require the data(i.e., the foregoing elements) necessary to perform the FFT be residentin a memory local to a node.

[0006] In a multidimensional FFT, N elements of a multidimensional arrayare distributed in a plurality of dimensions across nodes of adistributed-memory parallel multi-node computer. Many applications thatexecute on distributed-memory parallel multi-node computers spend alarge fraction of their execution time on calculating themultidimensional FFT. Since a motivation for the distributed-memoryparallel multi-node computers is faster execution, fast calculation ofthe multidimensional FFT for the distributed array is of criticalimportance. The N elements of the array are initially distributed acrossthe nodes in some arbitrary fashion particular to an application. Tocalculate the multidimensional FFT, the array of elements is thenredistributed such that a portion of the array on each node consists ofa complete row of elements in the x-dimension. A one-dimensional FFT oneach row in the x-dimension on each node is then performed. Since therow is local to a node and since each one-dimensional FFT on each row isindependent of the others, the one-dimensional FFT performed on eachnode requires no communication with any other node and may be performedusing abovementioned library routines. After the one-dimensional FFT,the array elements are re-distributed such that a portion of the arrayon each node consists of a complete row in the y-dimension. Thereafter,a one-dimensional FFT on each row in the y-dimension on each node isperformed. If there are more than two dimensions for the array, then there-distribution and a one-dimensional FFT are repeated for eachsuccessive dimension of the array beyond the x-dimension and they-dimension. The resulting array may be re-distributed into somearbitrary fashion particular to the application.

[0007] The treatment of the x-dimension and the y-dimension in sequenceis not fundamental to the multidimensional FFT. Instead, the dimensionsof the array may be treated in any order. For some applications or somecomputers, some orders may take advantage of some efficiency and thushave a faster execution than other orders. For example, the initialdistribution of the array across the nodes, which is in some arbitraryfashion particular to the application, may coincide with thedistribution necessary for the one-dimensional FFTs in the y-dimension.In this case, it may be fastest for the multidimensional FFT to treatthe y-dimension first, before treating the x-dimension and any otherremaining dimensions.

[0008] In the implementation of the multidimensional FFT describedabove, each re-distribution of the array between the one-dimensionalFFTs is an example of an “all-to-all” communication or re-distribution.In the all-to-all re-distribution, each node of the distributed-memoryparallel multi-node computer sends unique data (i.e., elements of thearray) to all other nodes utilizing a plurality of packets. Asabove-mentioned, fast calculation of the multidimensional FFT on thedistributed-memory parallel multi-node computer, is of criticalimportance. In the implementation described above, typically a largefraction of the execution time is spent to re-distribute the arrayacross the nodes the distributed-memory parallel multi-node computer.More particularly, a large fraction of execution time is spent on the“all-to-all” re-distribution of elements of the array across the nodesthe distributed-memory parallel multi-node computer.

[0009] Therefore there is a need in the art for providing a system andmethod for efficiently implementing the multidimensional FFT on the onthe distributed-memory parallel supercomputer. In particular, there is aneed in the art for providing a system and method for efficientlyimplementing the “all-to-all” re-distribution on the distributed-memoryparallel supercomputer for efficiently implementing the multidimensionalFFT.

SUMMARY OF THE INVENTION

[0010] It is therefore an object of the present invention to provide asystem and method for efficiently implementing the multidimensional FFTon an array distributed on a distributed-memory parallel supercomputer.

[0011] It is another object of the present invention to provide a systemand method for efficiently implementing the multidimensional FFT on thearray by efficiently implementing the “all-to-all” re-distribution onthe distributed-memory parallel supercomputer.

[0012] It is yet another object of the present invention to provide asystem and method for efficiently implementing the “all-to-all”re-distribution in applications other than the multidimensional FFT onthe distributed-memory parallel supercomputer.

[0013] According to an embodiment of the present invention, there isprovided a method for efficiently implementing a multidimensional FastFourier Transform (FFT) of a multidimensional array comprising aplurality of elements initially distributed in a multi-node computersystem comprising a plurality of nodes in communication over a network,the method comprising: distributing the plurality of elements of thearray in a first dimension across the plurality of nodes of the computersystem over the network to facilitate a first one-dimensional FFT;performing the first one-dimensional FFT on the elements of the arraydistributed at each node in the first dimension; re-distributing theone-dimensional FFT-transformed elements at each node in a seconddimension via “all-to-all” distribution in random order across othernodes of the computer system over the network; and performing a secondone-dimensional FFT on elements of the array re-distributed at each nodein the second dimension, wherein the random order facilitates efficientutilization of the network thereby efficiently implementing themultidimensional FFT.

[0014] According to another embodiment of the present invention, thereis provided a system for for efficiently implementing a multidimensionalFast Fourier Transform (FFT) of a multidimensional array comprising aplurality of elements initially distributed in a multi-node computersystem comprising a plurality of nodes in communication over a network,the system comprising: means for distributing the plurality of elementsof the array in a first dimension across the plurality of nodes of thecomputer system over the network to facilitate a first one-dimensionalFFT; means for performing the first one-dimensional FFT on the elementsof the array distributed at each node in the first dimension; means forre-distributing the one-dimensional FFT-transformed elements at eachnode in a second dimension via “all-to-all” distribution in random orderacross other nodes of the computer system over the network; and meansfor performing a second one-dimensional FFT on elements of the arrayre-distributed at each node in the second dimension, wherein the randomorder facilitates efficient utilization of the network therebyefficiently implementing the multidimensional FFT.

[0015] According to yet another embodiment of the present invention,there is provided a program storage device, tangibly embodying a programof instructions executable by a machine to perform a method forefficiently implementing a multidimensional Fast Fourier Transform (FFT)of a multidimensional array comprising a plurality of elements initiallydistributed in a multi-node computer system comprising a plurality ofnodes in communication over a network, the method comprising:distributing the plurality of elements of the array in a first dimensionacross the plurality of nodes of the computer system over the network tofacilitate a first one-dimensional FFT; performing the firstone-dimensional FFT on the elements of the array distributed at eachnode in the first dimension; re-distributing the one-dimensionalFFT-transformed elements at each node in a second dimension via“all-to-all” distribution in random order across other nodes of thecomputer system over the network; and performing a secondone-dimensional FFT on elements of the array re-distributed at each nodein the second dimension, wherein the random order facilitates efficientutilization of the network thereby efficiently implementing themultidimensional FFT.

[0016] According to a further embodiment of the present invention, thereis provided a method for efficiently re-distributing a multidimensionalarray comprising a plurality of elements initially distributed in amulti-node computer system comprising a plurality of nodes incommunication over a network, the method comprising re-distributing theelements at each node via “all-to-all” distribution in random orderacross other nodes of the computer system over the network, wherein therandom order facilitates efficient utilization of the network.

[0017] According to yet a further embodiment of the present invention,there is provided a system for efficiently re-distributing amultidimensional array comprising a plurality of elements initiallydistributed in a multi-node computer system comprising a plurality ofnodes in communication over a network, the system comprising a means forre-distributing the elements at each node via “all-to-all” distributionin random order across other nodes of the computer system over thenetwork, wherein the random order facilitates efficient utilization ofthe network.

[0018] According to still a further embodiment of the present invention,there is provided a program storage device, tangibly embodying a programof instructions executable by a machine to perform a method forefficiently re-distributing a multidimensional array comprising aplurality of elements initially distributed in a multi-node computersystem comprising a plurality of nodes in communication over a network,the method comprising re-distributing the elements at each node via“all-to-all” distribution in random order across other nodes of thecomputer system over the network, wherein the random order facilitatesefficient utilization of the network.

BRIEF DESCRIPTION OF THE DRAWINGS

[0019] The objects, features and advantages of the present inventionwill become apparent to one skilled in the art, in view of the followingdetailed description taken in combination with the attached drawings, inwhich:

[0020]FIG. 1 illustrates an exemplary distributed-memory parallelsupercomputer that includes 9 nodes interconnected via amultidimensional grid utilizing a 2-dimensional 3×3 Torus networkaccording to the present invention;

[0021]FIG. 2 illustrates a more detailed representation of an exemplarynode from the distributed-memory parallel supercomputer of FIG. 1according to the present invention;

[0022]FIG. 3 illustrates an exemplary two-dimensional 9-row by 9-columnarray, which may efficiently be implemented for the multidimensional FFTaccording to the present invention;

[0023]FIG. 4 illustrates an exemplary distribution the two-dimensionalarray of FIG. 3 across the nodes of the supercomputer in FIG. 1according to the present invention;

[0024]FIG. 5 illustrates an exemplary first one-dimensional FFT of thetwo-dimensional array distributed across the nodes of the supercomputerof FIG. 1 according to the present invention;

[0025]FIG. 6 illustrates an exemplary re-distribution of a resultanttwo-dimensional array after the first one-dimensional FFT of FIG. 5according to the present invention;

[0026]FIG. 7 illustrates an exemplary second one-dimensional FFT of there-distributed array of FIG. 6 according to the present invention;

[0027]FIG. 8 illustrates an exemplary method flowchart depicting theimplementation of the two-dimensional FFT illustrated in FIGS. 4-7according to the present invention;

[0028]FIG. 9 illustrates an exemplary method flowchart that depicts thefilling of output queues on the exemplary node with packets destined forother nodes on the distributed-memory parallel supercomputer accordingto the present invention; and

[0029]FIG. 10 illustrates an exemplary method flowchart that depicts howthe packets in the output queues on the exemplary node are drained intoinjection FIFOs for subsequent insertion on the Torus network 100according to the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT OF THE INVENTION

[0030] The present invention is directed to a system and method forefficiently implementing the multidimensional Fast Fourier Transform(i.e., “FFT”) on the distributed-memory parallel supercomputer. Moreparticularly, the present invention implements an efficient “all-to-all”re-distribution of elements distributed at nodes of thedistributed-memory parallel supercomputer to achieve an efficientimplementation of the multidimensional FFT.

[0031] According to the present invention, the FFT is implemented on thedistributed-memory parallel supercomputer, as a series ofone-dimensional transforms, which require one or more “all-to-all”re-distributions of a multidimensional array across the nodes of thedistributed-memory parallel supercomputer. The distributed-memoryparallel supercomputer utilizes a Torus-based network for theinterconnection of and communication between nodes of the supercomputer.As will be described below, each node implements a hardware router forefficiently routing packets that include elements of the array acrossthe nodes of the supercomputer interconnected via the Torus-basednetwork. Therefore, the present invention couples the implementation ofthe multidimensional FFT as a series of one-dimensional transforms ofthe multi-dimensional array with the foregoing hardware routing toobtain the efficient FFT implementation according to the presentinvention.

[0032] Further according to the present invention, thedistributed-memory parallel supercomputer comprises a plurality ofnodes, each of which includes at least one processor that operates on alocal memory. The nodes are interconnected as a multidimensional gridand they communicate via grid links. Without losing generality and inorder to make the description of this invention easily understandable toone skilled in the art, the multidimensional node grid of thesupercomputer will be described as an exemplary 2-dimensional grid.Notwithstanding the fact that only the 2-dimensional node grid isdescribed in the following description, it is contemplated within thescope of the present invention that node grids of other dimensions mayeasily be provided based on the teachings of the present invention. Itis noted that the distributed-memory parallel supercomputer may utilizea 3-dimensional or greater Torus-based architecture. Additionally,without losing generality and in order to make the description of thisinvention easily understandable to one skilled in the art, themultidimensional array used by the multidimensional FFT will bedescribed as an exemplary 2-dimensional array. Notwithstanding the factthat only the 2-dimensional array is described in the followingdescription, it is contemplated within the scope of the presentinvention that arrays of additional dimensions may easily be providedbased on the teachings of the present invention. It is further notedthat there is no correspondence between the number of dimensions in theTorus-based architecture and the number of dimensions in the array. Thearray must be of sufficient size such that it can be distributed acrossthe nodes or a subset of the nodes of the supercomputer for implementingthe multidimensional FFT according to the present invention.

[0033]FIG. 1 is an exemplary illustration of distributed-memory parallelsupercomputer that includes 9 nodes interconnected via amultidimensional grid utilizing a 2-dimensional 3×3 Torus network 100,according to the present invention. It is noted that the number of nodesis in exemplary fashion limited to 9 nodes for brevity and clarity, andthat the number of nodes may significantly vary depending on aparticular architectural requirements for the distributed-memoryparallel supercomputer. FIG. 1 depicts 9 nodes labeled as Q11-Q33, apair of which is interconnected by a grid link. In total, the 9-nodeTorus network 100 is interconnected by 18 grid links, where each node isdirectly interconnected to four other nodes in the Torus network 100 viaa respective grid link. It is noted that unlike a mesh, the exemplary2-dimensional Torus network 100 includes no edge nodes. For example,node Q11 is interconnected to node Q31 via grid link 102; to node Q13via grid link 104; to node Q21 via grid link 106; and finally to nodeQ12 via grid link 108. As another example, Node Q22 is interconnected toNode Q12 via grid link 110; to node Q21 via grid link 112; to node Q32via grid link 114 and finally to Node Q23 via grid link 116. Other nodesare interconnected in a similar fashion.

[0034] Further with reference to FIG. 1, data (i.e., elements of thearray) communicated between nodes is transported on the network in oneor more packets. For any given communication between a pair of nodes, aplurality of packets are required if the amount of data to becommunicated exceeds the packet-size supported by the Torus network 100.A packet comprises a packet header and the data carried by the packet.The packet header includes information required by the Torus network 100to transport the packet from a source node to a destination node. In thedistributed-memory parallel supercomputer of the present patentapplication, each node on the network is identified by a logical addressand the packet header includes a destination address so that the packetis automatically routed to a node on the network as identified by adestination.

[0035]FIG. 2 is a more detailed representation 200 of an exemplary node,e.g., node Q11, from the distributed-memory parallel supercomputer ofFIG. 1 according to the present invention. The node Q11 comprises atleast one processor 202 that operates on local memory 204. The nodefurther comprises a router 206 that routes, i.e., sends and receives,packets on the grid links 102,104,106 and 108, which connect the nodeQ11 to its neighboring nodes Q31, Q13, Q21 and Q12, respectively, asparticularly illustrated in FIG. 1. Yet further, the node comprises areception buffer 208 for buffering packets received by the router 206,which are destined for the local processor 202. The local processor 202may easily periodically poll the reception buffer 208 in order todetermine if there are packets in the reception buffer and then retrievethe packets that are buffered in the reception buffer 208. Depending ona particular application and the packets, the local processor 202 maywrite the contents of the packets into memory 204.

[0036] Further with reference to FIG. 2, the node Q11 comprises fourinjection First-In-First-Out (i.e., “FIFO”) buffers 810, which areparticularly labeled X+, X−, Y+ and Y−.

[0037] The processor places outbound packets into one or more outputqueues 212 of the local memory 2104, which store packets destined forother nodes until they can be placed into the injection FIFOs 210. Whileinjection FIFOs are not full, the processor places outbound packets intothe injection FIFOs 210. Upon a particular packet reaching the head ofan injection FIFO 210, the packet is removed from the injection FIFO 210by the router 206 and the router 206 inserts the packet onto a grid link102,104,106 and 108 toward a destination node for the particular packet.The four injection FIFOs 210 are treated equivalently by the router 206and by the hardware of the local processor 202.

[0038] Yet further with reference to FIG. 2, the router 206 comprisesseveral simultaneous routing characteristics. The routing firstrepresents virtual cut-through routing. For example, if an incomingpacket on one of the grid links is not destined for the local processor202 of node Q11, then the router 206 forwards the packet onto one of theoutgoing grid links 102, 104, 106 and 108. The router 206 performs theforwarding without the involving the local processor 202. The routingfurther represents shortest-path routing. For example, a packet sent bynode Q11 to node Q13 (See FIGS. 1 and 8) that travels over the grid link104 represents a shortest path route. Any other path would by longer. Asanother example, a packet sent by node Q11 to node Q22 may travel overgrid links 106 and 112 or alternatively over grid links 108 and 110.This type of routing is represents an adaptive type of routing. Thus,there may be a choice of grid links by which a packet may leave a nodein transit for another node over the Torus-based network 100. In theprevious example, the packet may leave the node Q11 via the grid link106 or 108. Adaptive routing allows the router 206 to choose the lessbusy outgoing grid link for a packet or to choose the outgoing grid linkbased on some other criteria. It is noted that the adaptive routing isnot just performed at the source node of a packet, e.g., node Q11, butis performed at each intermediate node that a packet cuts through on theway to the packet's destination node over the Torus-based network 100 ofFIG. 1. The description below with reference to FIGS. 9 and 10particularly describes how the present invention performs the foregoingrouting of packets across the nodes of the supercomputer over the Torusnetwork 100.

[0039]FIG. 3 is an exemplary two-dimensional 9-row by 9-column array 300that includes 81 elements, which may efficiently be implemented for themultidimensional FFT according to the present invention. It is notedthat the exemplary two-dimensional array 300 is easily extended to othertwo-dimensional arrays including a different number of rows and columns(e.g., 10-row by 11-column two-dimensional array), which may be utilizedfor implementing the FFT on the distributed-memory parallelsupercomputer according to the present invention. In the array 200, thefirst row of the array comprises elements A11, A12 . . . A19, while thefirst column of the array comprises elements A11, A21 . . . A 91.

[0040]FIG. 4 is an exemplary distribution illustration 400 of how thetwo-dimensional array 300 of FIG. 3 is distributed across the nodesQ11-Q33 in FIG. 1 according to the present invention. It is noted thatthe array may initially be distributed across the nodes in somearbitrary fashion that is particular to an application. According topresent invention, the array re-distributed such that a portion of thearray on each node Q11 . . . Q33 comprises the distribution illustratedin FIG. 4. This re-distribution is similar to that described below withreference to FIGS. 5 and 6. As particularly depicted in the distributionillustration 400, each node of FIG. 1 includes a portion of thetwo-dimensional array 300 of FIG. 3. For example, node Q11 comprises thefirst row of the array 300, i.e., elements A11, A12 . . . A19. Asanother example, node Q12 comprises the second row of the array 300,i.e., elements A21, A22 . . . A23. It is noted that other nodes Q13-Q33of FIG. 1 comprise respective rows 3 through 9 of array 300, asparticularly depicted in distribution illustration 400 of FIG. 4. Inexemplary distribution of FIG. 4, the assignment of a particular node toa particular row of array elements is not fundamental. Instead, it isnoted that any assignment is feasible. For various applications and/orcomputers, some assignments may take advantage of efficiencies offeredby the applications and/or computers and thus produce faster executionthan other assignments. For example, it may be that the fastest way toperform the multidimensional FFT may be to reverse the assignments ofnodes Q11 and Q12 from those illustrated in FIG. 4.

[0041]FIG. 5 is an exemplary illustration 500 that depicts a firstone-dimensional FFT on the two-dimensional array of FIG. 4 that wasdistributed across the nodes Q11-Q33 over the two-dimensional Torusnetwork 100 of FIG. 1. As particularly noted above, the multidimensionalFFT according to the present invention is accomplished by performing aseries of one-dimensional FFTs. Thus according to the present invention,the multi-dimensional FFT of the two-dimensional array 300 may beimplemented as a series of one-dimensional FFTs. Therefore, aone-dimensional FFT is performed on each row of elements distributed ateach node. For example, a one-dimensional FFT is performed for theelements distributed at node Q11, i.e., elements in the first row ofarray 300 that were distributed to node Q11. One-dimensional FFTs areperformed for elements (i.e., rows of elements) at each node Q12-Q33.The result is an array of elements transformed by the firstone-dimensional FFT. More particularly, the result of theone-dimensional FFT on each row at each node is a row of the same lengthas particularly illustrated in FIG. 5. For example, a one-dimensionalFFT performed on the first row at node Q11 of FIG. 4, which compriseselements A11, A12 . . . A19, results in a first row at node Q11 of FIG.5, which comprises elements B11, B12 . . . B19. Furthermore, theone-dimensional FFT performed on each row at each node is independent ofthe one-dimensional FFT performed on any other row at another node. Theparticular distribution of data illustrated in FIG. 4 enables each nodeto perform the one-dimensional FFT on the row of elements distributed atthat node, without communication with any other node on the Torusnetwork 100 of FIG. 1. Therefore, since no communication is requiredbetween the nodes, these one-dimensional FFTs are performed fast. It isnoted that at each node, in addition to the resulting row in FIG. 5, theoriginal row in FIG. 4 may continue to exist and be of interest for aparticular application, but the original row is no longer needed for thesecond one-dimensional FFT in the series of FFTs required for themultidimensional FFT according to the present invention, as particularlyillustrated in FIGS. 6 and 7.

[0042]FIG. 6 is an exemplary “all-to-all” re-distribution illustration600 that depicts how each resulting row of elements transformed via thefirst-dimension FFT of FIG. 5 is re-distributed across the nodes Q11-Q33for performing the second-dimension FFT according to the presentinvention. More particularly, each resulting row of elements that isdistributed at each node Q11 . . . Q33 of FIG. 5 is re-distributed overthe Torus network 100 so that each successive node receives a successivecolumn of elements as particularly depicted in FIG. 6. This efficientre-distribution is the “all-to-all” re-distribution, which enables anefficient implementation of the multidimensional FFT on thedistributed-memory parallel supercomputer according to the presentinvention. For example, the first node Q11 receives the first column ofelements, i.e., first elements from each of the nodes Q11 . . . Q33. Asanother example, node Q12 receives the second column of elements, i.e.,second elements from each of the nodes Q11 . . . Q33. Thisredistribution is performed for each column in FIG. 5. In exemplaryre-distribution of FIG. 6, the assignment of a particular node to aparticular row of array elements is not fundamental. Instead, it isnoted that any assignment is feasible. For various applications and/orcomputers, some assignments may take advantage of efficiencies offeredby the applications and/or computers and thus produce faster executionthan other assignments. For example, the fastest way to perform themultidimensional FFT may be to reverse the assignments of nodes Q11 andQ12 from those illustrated in FIG. 6. The description below withreference to FIGS. 9 and 10 particularly describes how the presentinvention performs the “all-to-all” re-distribution of array elementsacross the nodes of the supercomputer over the Torus network 100. The“all-to-all” re-distribution of the elements at each node Q11 . . . Q33is fast since it takes advantages of the communication characteristicsof the Torus network 100. In the re-distribution illustrated in FIG. 6,each node from Q11 . . . Q33 nodes sends a single array element to everyother node. The following description assumes that each element of thearray is a quantity of data larger than the quantity of data carried bya single packet. Thus, a plurality of packets is needed to transmit eachelement of the array to a destination node over the Torus network 100.This closely resembles the typical real-world re-distribution, where dueto much larger array sizes, each node sends many array elements to everyother node, typically requiring many packets.

[0043]FIG. 7 is an exemplary illustration 700 that depicts a secondone-dimensional FFT on the two-dimensional array of FIG. 6 that wasredistributed across the nodes Q11-Q33 over the two-dimensional Torusnetwork 100 of FIG. 1 according to the present invention. Asparticularly noted above, the multidimensional FFT according to thepresent invention is accomplished by performing a series ofone-dimensional FFTs, where FIG. 7 depicts the second one-dimensionalFFT in that series according to the present invention. Therefore, aone-dimensional FFT is performed on the column of elements that weredistributed to each node as illustrated in FIG. 5. For example, aone-dimensional FFT is performed for the elements distributed at nodeQ11, i.e., elements B11, B21 . . . B91 in FIG. 6 that were distributedas a row to node Q11 form the first column of FIG. 5. Additionally,one-dimensional FFTs are performed on rows of elements (i.e.,distributed from successive columns of elements of FIG. 5) at each nodeQ12-Q33. The result of the one-dimensional FFT on each row is a row ofthe same length as particularly illustrated in FIG. 7. For example, aone-dimensional FFT performed on the first row at node Q11 of FIG. 6,which comprises elements B11, B21 . . . A91, results in a first row atnode Q11 of FIG. 7, which comprises elements C11, C21 . . . C91. Asmentioned above with regard to the first FFT, the one-dimensional FFTperformed on each row at each node is independent of the one-dimensionalFFT performed on any other row at another node. The particulardistribution of data illustrated in FIG. 6 enables each node to performthe one-dimensional FFT on the row of elements distributed at that node,without communication with any other node on the Torus network 100 ofFIG. 1. Therefore, since no communication is required between the nodes,these one-dimensional FFTs are performed fast.

[0044]FIG. 8 is an exemplary method flowchart that illustrates theimplementation of the two-dimensional FFT of an array on the distributeddistributed-memory parallel supercomputer of FIG. 1 that utilizes a2-dimensional Torus network 100 for communication between nodes Q11 . .. Q33 of the supercomputer. In the following description, FIG. 8 is willbe described on the basis of FIGS. 1-7 for efficiently performing thetwo-dimensional FFT. At step 802, the multi-dimensional FFT of atwo-dimensional array illustrated in FIG. 3 in the distributeddistributed-memory parallel supercomputer of FIG. 1 is started. It isnoted that at step 702, the array illustrated in FIG. 3 is distributedacross the nodes in some arbitrary fashion that may be particular to anapplication. At step 804, elements (i.e., the data) of the array 300 areefficiently re-distributed across nodes Q11 . . . Q33, as particularlyillustrated in FIG. 4. At step 806, each node performs a firstone-dimensional FFT (out of a series of one-dimensional FFTs) on a rowof elements of the array stored at that node, as illustrated in FIG. 4,and the result particularly illustrated in FIG. 5. As described withregard to FIGS. 5 and 6, columns of one-dimensional FFT-transformedelements are re-distributed across the nodes Q11 . . . Q33 of thesupercomputer utilizing the Torus-based architecture of FIG. 1 at step808. At step 810, each node performs a second one-dimensional FFT on asuccessive column of a first one-dimensional FFT-transformed elementsillustrated of FIG. 6 that is distributed as a row of elements in FIG.6. The result of the second one-dimensional FFT is illustrated in FIG.7. At step 812, the multi-dimensional FFT of the two-dimensional arrayillustrated in FIG. 3 in the supercomputer of FIG. 1 is ended. Asparticularly described above, between the two one-dimensional FFTs thereis a fast re-distribution of elements across the nodes Q11 . . . Q33.

[0045] The above-described multidimensional FFT on an array of elementsdistributed across nodes of a distributed-memory parallel supercomputercoupled with redistribution of the elements across the nodes areillustrative of the invention. More particularly, the present inventionutilizes efficient hardware routing of the Torus-based architecturecoupled with a series of one-dimensional FFTs to achieve an efficientimplementation of the multidimensional FFT on the distributed-memoryparallel supercomputer. As noted above, the teachings according to thepresent invention may be utilized for performing efficientmultidimensional FFTs in other number of array dimensions, in otherarray sizes, and in other number of Torus network dimensions, e.g.,3-dimensional Torus. Additionally, the teachings according to thepresent invention may be utilized for performing “all-to-all”communication between nodes of the distributed-memory parallelsupercomputer on a Torus network of arbitrary dimensions.

[0046]FIG. 9 is an exemplary method flowchart 900 that depicts thefilling of one or more output queues 212 on an exemplary node Q11 ofFIG. 2 with packets destined for other nodes, e.g., nodes Q22 and Q33,on the distributed-memory parallel supercomputer according to thepresent invention. The “all-to-all” re-distribution illustrated in FIG.6 above is implemented as follows according to the present invention.Assume that Qxy denotes a generic node (e.g., node Q11) with anx-coordinate value x and a y-coordinate value y (e.g., x=1; y=1). Thus,according to the “all-to-all” re-distribution, node Qxy (e.g., node Q11)needs to send a plurality of total packets (i.e., k packets) to everynode Qab for all possible values of a and b (e.g., Q12, Q13; Q21, Q22,Q23; and Q31, Q32, Q33 as illustrated in FIG. 1; it is noted that Q11does not send packets to itself). To perform the re-distribution as fastas possible, the grid links of the Torus network 100 must efficientlyutilized. If packets are not scheduled in an efficient order, then thegrid link utilization may be very inefficient. For example, if everynode first sends packets only in the positive X+ direction, then all thegrid links in the negative X− direction will be idle, hence there-distribution will not be performed as fast as possible and themultifield FFT will not be implemented as efficiently as possible.According to the present invention, the fast re-distribution takesadvantage of the adaptive routing capability of the Torus-based network100 such that packet scheduling is implemented efficiently, asparticularly illustrated below.

[0047] Thus with reference to FIG. 9, there are Nx*Ny nodesinterconnected by the Torus network 100 (i.e., 3×3=9 nodes in FIG. 1)that need to exchange packets, which include elements of thetwo-dimensional array. At step 902, the exemplary method starts. At step904, at each node Q11 . . . Q33 there is created an array (i.e.,random₁₃ map[ ] array) that assigns each node on the Torus network 100 aunique number between 0, . . . , Nx * Ny−2. Since a node does not sendpackets to itself, the total number of nodes that exchange packets are 0to Nx*Ny−2. It is noted that the assignments at step 904 are generatedrandomly. At this point, assume that the total number of packets that anode requires to send an element of the array to another node is kpackets (e.g., 6 packets). Thereafter, assume that total k packets=diterations * b packets, where d is the number of iterations necessaryand transmit b packets per iteration for a total number of k packets. Itis noted that b may be chosen as necessary for efficiency and maylikewise be equal to 1. For example, to transmit a total of 6 packets,it can be chosen to transmit 2 packets per iteration on each of 3iterations for the total of 6 packets. Therefore, at step 906, a loop isinitiated for id from 1 to d iterations. At step 908, a queue counter isinitialized to zero. It is assumed that there are L output queues 212 (Lbeing greater than or equal to 1) for storing packets (or shortdescriptors of the packets such that the actual packet need not becopied), and all packets (or descriptors of the packets) for a givendestination will be placed into the same output queue. A particularoutput queue iL is selected in round-robin order at step 912 withinnested loops of FIG. 9. At step 910, a loop is initialized for iN valuefrom node 0 to node Nx*Ny−2, as an index into the array (i.e.,random_array[ ]) created at step 904. As the array created in step 904is indexed for a particular iN value, a random node value is obtainedfrom the random_array. At step 912, a first queue is selected inround-robin order. At step 914, a loop is initialized for ib from 1 to bpackets per d iterations. Subsequently, as steps 914 and 916, aplurality of b packets (e.g., b=2 packets from above example) destinedfor a given random node iN are added to the same output queue iL aspacket[node, id, ib]. At step 918, once all d iterations have beencompleted, the method ends. In sum with reference to the flowchart 900,during one d iteration a particular node “i” (e.g., processor 202 onnode Q11 of FIG. 2) will first place b number of packets that includedata for an element of the array destined for a node Modulus (i+1,Nx*Ny−1) in a first output queue, then particular node “i” will place bpackets that include data for an element of the array destined for anode Modulus (i+2, Nx*Ny−1) into a next output queue, and so on untilreaching node Modulus (i+(Nx*Ny−1), Nx*Ny−1). When the packets b packetshave been inserted for a given iteration into the output queues, thisprocess is repeated until the d iterations have all been completed. Theforegoing re-distribution achieves extremely high grid link utilizationon the Torus network 100 of FIG. 1, thereby efficiently implementing themultidimensional FFT according to the present invention.

[0048]FIG. 10 is an exemplary method flowchart 1000 that depicts how thepackets in the one or more output queues 212 on the exemplary node Q11of FIG. 2 are drained into the injection FIFOs 210 for subsequentinsertion on the Torus network 100 according to the present invention.Before describing FIG. 10 in detail, it is noted that the filling ofFIG. 9 and the draining of FIG. 10 may be performed concurrently withone another. At step 1002, the exemplary method starts. At step 1004 itis determined whether all L output queues 212 are empty. At step 1006 aloop is initiated for iL from 1 to L, to iterate over all L outputqueues from. At step 1008 it is determined whether a particular outputqueue iL is empty. If the output queue iL is empty, the method continuesto the next iL output queue at step 1006. Otherwise, at step 1010, for apacket at the head of the output queue iL, possible directions forrouting the packet over the Torus network 100 are obtained. For examplewith reference to FIG. 1, assume that node Q11 placed a packet destinedto node Q22 into an output queue iL. The packet may travel from node Q11in the X+ direction (over grid link 108) followed by Y− direction (overgrid link 110) to reach node Q22, or it may travel in the Y− direction(over grid link 106) followed by the X+ direction (over grid link 112)to reach node Q22. Now back to FIG. 10, at step 1012 it is furtherdetermined whether all FIFOs 210 of FIG. 2 in the possible directionsfor the packet are full. As described above, each injection FIFO 210 hasa logical direction (e.g., X+) associated with it, which represents thatany packet placed in the injection FIFO 210 can move in the associatedlogical direction (e.g., X+ direction). If the injection FIFOs 210 forpacket directions are full, then the method skips the current outputqueue and continues by iterating to the next output queue at step 1006.Otherwise, at step 1014, the packet is moved from the output queue to aleast full FIFO 212 in one of the possible directions for that packet.It is noted that packets are removed from output the queues in around-robin order for insertion into the injection FIFOs 210 illustratedin FIG. 2. After the packet is moved, the method continues at step 1008for a next available packet in that output queue. Once all output queuesare empty, the method ends at step 1016.

[0049] In order to more fully demonstrate FIGS. 9 and 10, which describethe “all-to-all” routing, assume that the row of elements at node Q11 inFIG. 5, i.e., elements B11, B12 . . . B19, are to be re-distributedacross nodes Q12 . . . Q33 as illustrated in FIG. 6 over the Torusnetwork 100. Assume that the random mapping of nodes has followingvalues in random_map array={Q32; Q22; Q13; Q21; Q23; Q33; Q12; and Q31}.Therefore, the order of the array elements and their destination nodesfrom node Q11 is as follows: {B12 to Q12; B13 to Q13; B14 to Q21; B15 toQ22; B16 to Q23; B17 to Q31; B18 to Q32 and B19 to Q33}. The arrayelements are placed into the FIFOs 210 of node Q11 as follows: {B18 toQ32 via X+ or Y−; B15 to Q22 via X+ or Y+; B13 to Q13 via X−; B14 to Q21via Y+; B16 to Q23 via Y+ or X−; B19 to Q33 via X− or Y−; B12 to Q12 viaX+; and B17 to Q31 via Y−}. Thus for example, the FIFOs 210 on node Q11might be filled as illustrated in the table 1 below. TABLE 1 X+ X− Y+ Y−B18 to Q32 B13 to Q13 B15 to Q22 B14 to Q21 B12 to Q12 B19 to Q33 B16 toQ23 B17 to Q31

[0050] Notwithstanding the fact that the number of injection FIFOs wasdescribed above as equal to the number of grid links to a node (e.g., 4FIFOs and 4 grid links), the use of an injection FIFO that is restrictedto at least a particular grid link also is well-suited when number ofinjection FIFOs is not equal to the number of grid links. For example,if there are fewer injection FIFOs than grid links, then the use of abuffer may be restricted to at least one of several particular gridlinks. For another example, if there are more injection FIFOs than gridlinks, then there may be several injection FIFOs whose use is restrictedto at least the same particular grid link.

[0051] Although the implementation of the array re-distribution wasdescribed above with reference to efficient implementation of themultidimensional FFT, the “all-to-all” re-distribution is also wellsuited for any type of array re-distributions over the Torus network 100of FIG. 1.

[0052] While the invention has been particularly shown and describedwith regard to preferred embodiments thereof, it will be understood bythose skilled in the art that the foregoing and other changes in formand details may be made therein without departing from the spirit andscope of the invention.

Having thus described our invention, what we claim as new, and desire tosecure by Letters Patent is:
 1. A method for efficiently implementing amultidimensional Fast Fourier Transform (FFT) of a multidimensionalarray comprising a plurality of elements initially distributed in amulti-node computer system comprising a plurality of nodes incommunication over a network, the method comprising: (a) distributingthe plurality of elements of the array in a first dimension across theplurality of nodes of the computer system over the network to facilitatea first one-dimensional FFT; (b) performing the first one-dimensionalFFT on the elements of the array distributed at each node in the firstdimension; (c) re-distributing the one-dimensional FFT-transformedelements at each node in a second dimension via “all-to-all”distribution in random order across other nodes of the computer systemover the network; and (d) performing a second one-dimensional FFT onelements of the array re-distributed at each node in the seconddimension, wherein the random order facilitates efficient utilization ofthe network thereby efficiently implementing the multidimensional FFT.2. The method for efficiently implementing a multidimensional FFTaccording to claim 1, wherein the method further comprises the step of:re-distributing the elements of the array at each node in a thirddimension via the “all-to-all” distribution in random order across othernodes of the computer system over the network; performing aone-dimensional FFT on elements of the array re-distributed at each nodein the third dimension; and repeating the steps of re-distributing theelements of the array in random order across nodes and performing theone-dimensional FFT on the re-distributed elements at each node forsubsequent dimensions.
 3. The method for efficiently implementing amultidimensional FFT according to claim 1, wherein the method comprisesa step of generating a random order of other nodes for re-distributingthe one-dimensional FFT-transformed elements at each node.
 4. The methodfor efficiently implementing a multidimensional FFT according to claim3, wherein each of the plurality of elements is re-distributed betweennodes of the computer system via a plurality of total packets.
 5. Themethod for efficiently implementing a multidimensional FFT according toclaim 4, wherein the method further comprises the steps of: providing aplurality of output queues at each node; iterating thru the other nodesin generated random order a plurality of times; and outputting to anoutput queue for each other node at least one packet of the plurality oftotal packets during each iteration.
 6. The method for efficientlyimplementing a multidimensional FFT according to claim 5, wherein themethod further comprises the steps of: providing a plurality ofinjection first-in-first-out (FIFO) buffers, each FIFO buffer fortransmitting packets in at least a particular direction on the network;iterating through the plurality of output queues at a node to identify apacket at the head of each queue; obtaining possible routing directionsassociated with the packet at the head of each queue; and moving thepacket from the head of each queue to a least full FIFO buffer in one ofthe possible routing directions associated with the packet.
 7. A systemfor efficiently implementing a multidimensional Fast Fourier Transform(FFT) of a multidimensional array comprising a plurality of elementsinitially distributed in a multi-node computer system comprising aplurality of nodes in communication over a network, the systemcomprising: (a) means for distributing the plurality of elements of thearray in a first dimension across the plurality of nodes of the computersystem over the network to facilitate a first one-dimensional FFT; (b)means for performing the first one-dimensional FFT on the elements ofthe array distributed at each node in the first dimension; (c) means forre-distributing the one-dimensional FFT-transformed elements at eachnode in a second dimension via “all-to-all” distribution in random orderacross other nodes of the computer system over the network; and (d)means for performing a second one-dimensional FFT on elements of thearray re-distributed at each node in the second dimension, wherein therandom order facilitates efficient utilization of the network therebyefficiently implementing the multidimensional FFT.
 8. The system forefficiently implementing a multidimensional FFT according to claim 7,wherein the method further comprises the step of: means forre-distributing the elements of the array at each node in a thirddimension via the “all-to-all” distribution in random order across othernodes of the computer system over the network; means for performing aone-dimensional FFT on elements of the array re-distributed at each nodein the third dimension; and means for repeating the steps ofre-distributing the elements of the array in random order across nodesand performing the one-dimensional FFT on the re-distributed elements ateach node for subsequent dimensions.
 9. The system for efficientlyimplementing a multidimensional FFT according to claim 7, wherein thesystems comprises a means for generating a random order of other nodesfor re-distributing the one-dimensional FFT-transformed elements at eachnode.
 10. The system for efficiently implementing a multidimensional FFTaccording to claim 9, wherein each of the plurality of elements isre-distributed between nodes of the computer system via a plurality oftotal packets.
 11. The system for efficiently implementing amultidimensional FFT according to claim 10, wherein the method furthercomprises the steps of: means for providing a plurality of output queuesat each node; means for iterating thru the other nodes in generatedrandom order a plurality of times; and means for outputting to an outputqueue for each other node at least one packet of the plurality of totalpackets during each iteration.
 12. The system for efficientlyimplementing a multidimensional FFT according to claim 11, wherein themethod further comprises the steps of: means for providing a pluralityof injection first-in-first-out (FIFO) buffers, each FIFO buffer fortransmitting packets in at least a particular direction on the network;means for iterating through the plurality of output queues at a node toidentify a packet at the head of each queue; means for obtainingpossible routing directions associated with the packet at the head ofeach queue; and means for moving the packet from the head of each queueto a least full FIFO buffer in one of the possible routing directionsassociated with the packet.
 13. A program storage device, tangiblyembodying a program of instructions executable by a machine to perform amethod for efficiently implementing a multidimensional Fast FourierTransform (FFT) of a multidimensional array comprising a plurality ofelements initially distributed in a multi-node computer systemcomprising a plurality of nodes in communication over a network, themethod comprising: (a) distributing the plurality of elements of thearray in a first dimension across the plurality of nodes of the computersystem over the network to facilitate a first one-dimensional FFT; (b)performing the first one-dimensional FFT on the elements of the arraydistributed at each node in the first dimension; (c) re-distributing theone-dimensional FFT-transformed elements at each node in a seconddimension via “all-to-all” distribution in random order across othernodes of the computer system over the network; and (d) performing asecond one-dimensional FFT on elements of the array re-distributed ateach node in the second dimension, wherein the random order facilitatesefficient utilization of the network thereby efficiently implementingthe multidimensional FFT.
 14. The program storage device for efficientlyimplementing a multidimensional FFT according to claim 13, wherein themethod further comprises the step of: re-distributing the elements ofthe array at each node in a third dimension via the “all-to-all”distribution in random order across other nodes of the computer systemover the network; performing a one-dimensional FFT on elements of thearray re-distributed at each node in the third dimension; and repeatingthe steps of re-distributing the elements of the array in random orderacross nodes and performing the one-dimensional FFT on there-distributed elements at each node for subsequent dimensions.
 15. Theprogram storage device for efficiently implementing a multidimensionalFFT according to claim 13, wherein the method comprises a step ofgenerating a random order of other nodes for re-distributing theone-dimensional FFT-transformed elements at each node.
 16. The programstorage device for efficiently implementing a multidimensional FFTaccording to claim 15, wherein each of the plurality of elements isre-distributed between nodes of the computer system via a plurality oftotal packets.
 17. The program storage device for efficientlyimplementing a multidimensional FFT according to claim 16, wherein themethod further comprises the steps of: providing a plurality of outputqueues at each node; iterating thru the other nodes in generated randomorder a plurality of times; and outputting to an output queue for eachother node at least one packet of the plurality of total packets duringeach iteration.
 18. The program storage device for efficientlyimplementing a multidimensional FFT according to claim 17, wherein themethod further comprises the steps of: providing a plurality ofinjection first-in-first-out (FIFO) buffers, each FIFO buffer fortransmitting packets in at least a particular direction on the network;iterating through the plurality of output queues at a node to identify apacket at the head of each queue; obtaining possible routing directionsassociated with the packet at the head of each queue; and moving thepacket from the head of each queue to a least full FIFO buffer in one ofthe possible routing directions associated with the packet.
 19. A methodfor efficiently re-distributing a multidimensional array comprising aplurality of elements initially distributed in a multi-node computersystem comprising a plurality of nodes in communication over a network,the method comprising re-distributing the elements at each node via“all-to-all” distribution in random order across other nodes of thecomputer system over the network, wherein the random order facilitatesefficient utilization of the network.
 20. The method for efficientlyre-distributing a multidimensional array according to claim 19, whereinthe method comprises a step of generating a random order of other nodesfor re-distributing the elements at each node.
 21. The method forefficiently re-distributing a multidimensional array according to claim20, wherein each of the plurality of elements is re-distributed betweennodes of the computer system via a plurality of total packets.
 22. Themethod for efficiently re-distributing a multidimensional arrayaccording to claim 21, wherein the method further comprises the stepsof: providing a plurality of output queues at each node; iterating thruthe other nodes in generated random order a plurality of times; andoutputting to an output queue for each other node at least one packet ofthe plurality of total packets during each iteration.
 23. The method forefficiently redistributing a multidimensional array according to claim22, wherein the method further comprises the steps of: providing aplurality of injection first-in-first-out (FIFO) buffers, each FIFObuffer for transmitting packets in at least a particular direction onthe network; iterating through the plurality of output queues at a nodeto identify a packet at the head of each queue; obtaining possiblerouting directions associated with the packet at the head of each queue;and moving the packet from the head of each queue to a least full FIFObuffer in one of the possible routing directions associated with thepacket.
 24. A system for efficiently re-distributing a multidimensionalarray comprising a plurality of elements initially distributed in amulti-node computer system comprising a plurality of nodes incommunication over a network, the system comprising a means forre-distributing the elements at each node via “all-to-all” distributionin random order across other nodes of the computer system over thenetwork, wherein the random order facilitates efficient utilization ofthe network.
 25. The system for efficiently re-distributing amultidimensional array according to claim 24, wherein the methodcomprises a means for generating a random order of other nodes forre-distributing the elements at each node.
 26. The system forefficiently re-distributing a multidimensional array according to claim25, wherein each of the plurality of elements is re-distributed betweennodes of the computer system via a plurality of total packets.
 27. Thesystem for efficiently re-distributing a multidimensional arrayaccording to claim 26, wherein the system further comprises: means forproviding a plurality of output queues at each node; means for iteratingthru the other nodes in generated random order a plurality of times; andmeans for outputting to an output queue for each other node at least onepacket of the plurality of total packets during each iteration.
 28. Thesystem for efficiently re-distributing a multidimensional arrayaccording to claim 27, wherein the system further comprises: means forproviding a plurality of injection first-in-first-out (FIFO) buffers,each FIFO buffer for transmitting packets in at least a particulardirection on the network; means for iterating through the plurality ofoutput queues at a node to identify a packet at the head of each queue;means for obtaining possible routing directions associated with thepacket at the head of each queue; and moving the packet from the head ofeach queue to a least full FIFO buffer in one of the possible routingdirections associated with the packet.
 29. A program storage device,tangibly embodying a program of instructions executable by a machine toperform a method for efficiently re-distributing a multidimensionalarray comprising a plurality of elements initially distributed in amulti-node computer system comprising a plurality of nodes incommunication over a network, the method comprising re-distributing theelements at each node via “all-to-all” distribution in random orderacross other nodes of the computer system over the network, wherein therandom order facilitates efficient utilization of the network.
 30. Theprogram storage device for efficiently re-distributing amultidimensional array according to claim 29, wherein the methodcomprises a step of generating a random order of other nodes forre-distributing the elements at each node.
 31. The program storagedevice for efficiently re-distributing a multidimensional array 29,wherein each of the plurality of elements is re-distributed betweennodes of the computer system via a plurality of total packets.
 32. Theprogram storage device for efficiently re-distributing amultidimensional array according to claim 31, wherein the method furthercomprises the steps of: providing a plurality of output queues at eachnode; iterating thru the other nodes in generated random order aplurality of times; and outputting to an output queue for each othernode at least one packet of the plurality of total packets during eachiteration.
 33. The program storage device for efficientlyre-distributing a multidimensional array according to claim 32, whereinthe method further comprises the steps of: providing a plurality ofinjection first-in-first-out (FIFO) buffers, each FIFO buffer fortransmitting packets in at least a particular direction on the network;iterating through the plurality of output queues at a node to identify apacket at the head of each queue; obtaining possible routing directionsassociated with the packet at the head of each queue; and moving thepacket from the head of each queue to a least full FIFO buffer in one ofthe possible routing directions associated with the packet.